Polarization fields for dynamic light field display

ABSTRACT

In exemplary implementations of this invention, a flat screen device displays a 3D scene. The 3D display may be viewed by a person who is not wearing any special glasses. The flat screen device displays dynamically changing 3D imagery, with a refresh rate so fast that the device may be used for 3D movies or for interactive, 3D display. The flat screen device comprises a stack of LCD layers with two crossed polarization filters, one filter at each end of the stack. One or more processors control the voltage at each pixel of each LCD layer, in order to control the polarization state rotation induced in light at that pixel. The processor employs an algorithm that models each LCD layer as a spatially-controllable polarization rotator, rather than a conventional spatial light modulator that directly attenuates light. Color display is achieved using field sequential color illumination with monochromatic LCDs.

RELATED APPLICATIONS

This application is a non-provisional of, and claims the benefit of thefiling date of, U.S. Provisional Application Ser. No. 61/564,855, filedNov. 29, 2011, the entire disclosure of which is herein incorporated byreference.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with U.S. government support under Grant NumberHS-1116452, awarded by the National Science Foundation, GrantHR0011-10-c-0073, awarded by the Defense Advanced Research ProjectsAgency (DARPA), and Young Faculty Award N66001-10-1-4041, awarded byDARPA. The government has certain rights in this invention.

FIELD OF THE TECHNOLOGY

The present invention relates generally to polarization.

SUMMARY

In exemplary implementations of this invention, a flat screen deviceproduces a 3D display. The displayed 3D scene appears different atdifferent viewing angles, just as an actual 3D scene does.

The flat screen device creates the illusion of looking into an actual 3Dscene—without the viewer wearing special glasses.

In exemplary implementations of this invention, the flat screen devicedisplays dynamically changing 3D imagery. The refresh rate is so fastthat the device may be used for 3D movies or for an interactive, 3Ddisplay.

As used herein, a “polarization rotator” is a device configured tochange the polarization state of light that travels through the device.For example, a polarization rotator may comprise a layer of liquidcrystal between a pair of transparent electrodes.

In exemplary implementations of this invention: An illumination sourceprovides light that is transmitted through at least (i) a multi-layeredstack of polarization rotators and (ii) a polarizer which is inoptically in front of the stack. Each of the polarization rotators is aspatially addressable device—it can dynamically vary, on a pixel bypixel basis, polarization state rotations of light being transmittedthrough the device.

In exemplary implementations of this invention: One or more processorsperform an optimization calculation to compute an optimal set ofpolarization state rotations induced in light at respective pixels inthe polarization rotators. For each respective light ray in a set oflight rays, the optimization calculation includes computing a summationof changes to the polarization state rotation of the respective lightray that occur as the respective light ray travels through the stack ofpolarization rotators.

In some implementations of this invention, the optimization calculationalso includes computing, according to Malus' law, an intensity of therespective light ray, the intensity being as of when the respectivelight ray emerges from the polarizer. Malus' law includes a term that isa square of a sinusoidal function. The optimization calculation mayinclude solving for the argument of the sinusoidal function, using anapproximation based on only a single period of the sinusoidal function.

In exemplary implementations of this invention: The apparatus producesan automultiscopic display. The processors output control signals thatcontrol and optimize polarization state rotations induced in light atrespective pixels of the polarization rotators. As a result, the lightfield that emerges from the front polarizer is the best approximation ofa target light field. For example, if the apparatus is rendering a 3Dscene, the optimized polarization state rotations minimize the errorbetween (a) the light field that is outputted from the front polarizerand (b) the light field that would be emitted by the actual 3D scene.

In illustrative implementations of this invention: Each polarizationrotator comprises a layer of liquid crystal. The polarization rotatorsare configured to dynamically vary voltage applied at respective pixelsin order to control polarization rotation at the respective pixels.

In some implementations of this invention: (a) the optimizationcalculation may be a constrained linear least squares optimizationcalculation; (b) the one or more processors may be configured to employa SART technique when performing the optimization calculation; and (c)the optimization calculation may be a non-linear optimizationcalculation.

In exemplary implementations of this invention, the SART algorithm isemployed to render dynamically changing 3D imagery at interactive rates.

In exemplary implementations of this invention, computations of thepolarization state rotations may be performed in advance or in realtime. For example, computations may be performed in real time for aninteraction with a human user.

In some implementations of this invention, computations for frames in amovie are performed in advance, and data regarding the computed framesis stored in memory. This data may be played in order to render a 3Dmovie. The 3D movie may be based on captured multi-view or light fielddata, or may instead be based on synthetic multi-view or light fielddata. The stored data may comprise voltage values for controllingpolarization state rotations at respective pixels in order to producethe light fields of the frames.

A prototype of this invention comprises a stacked set of liquid crystalpanels with a single pair of crossed linear polarizers. One or moreprocessors control the voltage at each pixel of each liquid crystalpanel, in order to control the polarization state rotation at thatpixel. The processor employs an algorithm that models each LCD layer asa spatially-controllable polarization rotator, as opposed to aconventional spatial light modulator that directly attenuates light.

In this prototype, color display is achieved using field sequentialcolor illumination with monochromatic LCDs. Each of the polarizationrotators is monochromatic and the illumination source is a strobebacklight configured to sequentially illuminate the polarizationrotators with varying colors of light. Alternately, color display may beachieved by other technologies, including color filter arrays.

In this prototype, processors control the display, at interactiverefresh rates, by using a simultaneous algebraic reconstructiontechnique (SART) algorithm to tomographically solve for the optimalspatially-varying polarization state rotations applied by each layer.Modified off-the-shelf LCD panels are used as polarization rotators.

This prototype achieves increased brightness, higher resolution, andextended depth of field, as compared to conventional display methods forautomultiscopic dual-layer and multi-layer LCDs.

The description of the present invention in the Summary and Abstractsections hereof is just a summary. It is intended only to give a generalintroduction to some illustrative implementations of this invention. Itdoes not describe all of the details of this invention. This inventionmay be implemented in many other ways.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a prior art example of a stack of spatial lightattenuators.

FIG. 2 shows a stack of polarization rotators, in an exemplaryembodiment of this invention.

FIG. 3 is a diagram showing a light ray traveling through a stack ofpolarization rotators.

FIG. 4 is a flow chart for controlling a stack of polarization rotatorswith a linear optimization algorithm.

FIG. 5 is a flow chart for controlling a stack of polarization rotatorswith a non-linear optimization algorithm.

FIGS. 2-5 illustrate some illustrative implementations of thisinvention, or provide information that relates to those implementations.However, this invention may be implemented in many other ways. The aboveFIGS. 1-5 do not show all of the details of this invention.

DETAILED DESCRIPTION

FIG. 1 shows an example of a conventional multilayered LCD with four LCDlayers 101, 103, 105, 107. Each of these LCD layers comprises a thinliquid crystal layer, enclosed within glass sheets with embedded,two-dimensional electrode arrays. The four LCD layers 101, 103, 105, 107are sandwiched between five polarizers 111, 113, 115, 117, 119.Polarizers 111, 115, 119 are crossed with polarizers 113, 117. Abacklight 121 provides uniform illumination.

In the example shown in FIG. 1, each LCD layer (together with thepolarizers immediately adjacent to it) functions as a spatial lightattenuator.

FIG. 2 shows a stack of polarization rotators, in an exemplaryimplementation of this invention. Similar to FIG. 1, the stack comprisesfour LCD layers 201, 203, 205, 207 and a backlight 221 for uniformillumination. Each of these LCD layers comprises a thin liquid crystallayer, enclosed within glass sheets with embedded, two-dimensionalelectrode arrays.

Unlike FIG. 1, there are only two polarizers 211, 213 in FIG. 2, one ateach end of the stack. In FIG. 2, each of the LCD layers 201, 203, 205,207 functions as a polarization rotator, rather than as a spatialattenuator.

Each time that light passes through a polarizer, some of the light isnot transmitted (even if the polarized light and polarizer are aligned,and even more so if they are not aligned). Thus, generally speaking, themore polarizers in a stack, the more the light is attenuated as itpasses through the stack. Advantageously, a stack of polarizationrotators (as in FIG. 2) may have less polarizers than a stack of spatiallight attenuators (as in FIG. 1). Thus, a stack of polarization rotatorsmay let more light through (i.e., be more optically efficient) than astack of spatial attenuators.

In exemplary embodiments of this invention, color 3D displays areachieved by using field sequential color (FSC), rather than a colorfilter array. A strobed LED backlight successively illuminates ahigh-speed monochromatic LCD with varying color sources. If strobingoccurs faster than the human flicker fusion threshold, a color image isperceived. The color image can be dynamically changed.

In exemplary implementations of this invention, FSC is preferable to acolor filter array, because FSC tends to attenuate light less than acolor filter array.

In the example shown in FIG. 2, one or more processors 231 are employedto, among other things, process data, perform calculations, and outputcontrol signals. The control signals may, among other things, controlthe strobed LED backlight 221. Also, the control signals may control thevoltages applied at respective pixels of the polarization rotators 201,203, 205, 207, and thus may control the polarization state rotationsinduced in light being transmitted through the respective pixels.

The calculations by the one or more processors 231 may includeoptimizing these voltages in order to produce a light field that bestapproximates a target light field. These calculations may be performedin real time for GPU-based interactions with a human user. Furthermore,one or more memory storage units 233 may be used to store data. The datathat is stored may include pre-computed data that can be played in orderto render a 3D movie. One or more input devices 235 may be used toaccept input from a human user. The one or more processors 231 may beconnected by wired or wireless connections to other parts of the displayapparatus, including (a) the polarization rotators 201, 203, 205, 207,(b) the strobed LED backlight 221, (c) the one or more memory storageunits 233, and (d) the one or more input devices 235. The location ofthe processors, memory units, and input devices may vary, depending onthe particular implementation of this invention. For example, any one ormore of the processors, memory units and input devices may be housedwith, or located remotely from, the polarization rotators.

In exemplary implementations of this invention,computationally-efficient algorithms control multi-layered LCDs toproduce a dynamic, automultiscopic display. A stack of multiple liquidcrystal panels is covered with a single pair of crossed linearpolarizers, one at each end of the stack. Each liquid display panelfunctions as a polarization rotator, rather than as a light attenuator.An efficient tomographic solver enables interactive applications andpractical dynamic light field display. The polarization field displaysachieve increased brightness, higher resolution, and extended depth offield, compared to conventional automultiscopic multi-layer LCDs thatfunction as a stack of spatial light attenuators.

In exemplary implementations of this invention, an algorithm models apolarization-based light field display as a constrained linearleast-squares problem, and solves the problem using a simultaneousalgebraic reconstruction technique (SART), allowing dynamic light fielddisplay.

According to principles of this invention, SART may be used foroptimization for dynamic light field display for both polarization-basedand attenuation-based multi-layer LCD architectures.

A prototype of this invention employs field sequential color, a pair ofcrossed polarizers, and layered, monochromatic LCDs. The LCD layerscomprise modified off-the-shelf LCD panels. This prototype achievesinteractive display with dynamic imagery using a GPU-based SART solver.

In exemplary implementations, the GPU-based SART solver enable controlof either attenuation-based or polarization-based displays atinteractive refresh rates. The display may reproduce only a centralzone. In that case, tracking (e.g., gaze-tracking may be used for widerfields of view.

As is well known in the art, the transformation of polarized light dueto passage through layered materials may be modeled by Jones calculus.In this approach, orthogonal components of the electric field arerepresented as a complex-valued Jones vectors. The optical action of agiven element (e.g., a birefringent layer or polarizing film) isrepresented by a Jones matrix, with the product of this matrix and aJones vector encoding the polarization state transformation. Jonescalculus may be used to characterize polarization properties of LCDpanels. For example, analytic Jones matrices may characterize twistednematic (TN), vertical alignment (VA), and in-plane switching (IPS)panels.

In some implementations of this invention, processors employ asimplified Jones matrix model, wherein LCDs are approximated asspatially-controllable polarization rotators. This simplified matrixmodel assumes that light outputted from each LCD panel is linearlypolarized. However, conventional LCDs actually output light that isslightly elliptical. As a result, if the simplified matrix model is usedwith conventional LCD panels, then visual artifacts may appear in thedisplay. There are at least two solutions to this problem. In exemplaryimplementations of this invention, either or both of these two solutionsmay be used. First, the LCD panels may be engineered to output lightthat conforms to the simplified matrix model. For example, compensationfilms may be employed to correct the output of each LCD panel, so thatlight outputted from each panel is linearly polarized. Second, a moredetailed Jones matrix model of the LCD panels may be employed. A benefitof this second approach is that it can better model actual lighttransmission through the stack, and thus may reduce visible artifacts. Adrawback of this second approach is that it increases computationalcomplexity and thus may lead to decreased refresh rates.

In exemplary implementations, the liquid crystal is operated as avoltage-controlled wave plate to produce polarization state rotations.Layered constructions of such panels are ideally suited to implementpractical polarization field displays.

In exemplary implementations of this invention, applying a voltageacross an electrode pair (on either side of a liquid crystal layer)alters the polarization properties of a pixel. According to principlesof this invention, the effect can be approximated as inducing a rotationof the polarization state of light rays traversing the pixel. Thisapproximation holds to varying degrees of accuracy under differentactual conditions.

In exemplary implementations, the transmitted intensity I that passesthrough the front polarizer (at the end of the stack closest to theobserver) is modeled by Malus' law

I=I ₀ sin²(θ)   (Eq. 1)

where I₀ is the intensity after passing through the first polarizer andθ is angle of polarization after passing through the liquid crystallayer closest to the front polarizer, defined relative to the axis ofthe front polarizer. This model only strictly applies for rays orientedperpendicular to the display surface. At oblique angles, light leakageoccurs through crossed polarizers and birefringence of the liquidcrystal produces elliptical, rather than linear, polarization states.However, this model is a close approximation for the viewing anglesconsidered in the prototype.

In exemplary implementations of this invention, a 3D scene is displayed.A front polarizer adjacent to multi-layer LCDs outputs a light fieldthat approximates the light field that would be emitted by the actual 3Dscene.

In the example shown in FIG. 2 (an exemplary implementation of thisinvention) a backlight 221 is covered by multiple, disjoint spatiallight modulators 201, 203, 205, 207. To maximize optical efficiency,field sequential color illumination is used; this eliminates K layers ofcolor filters that would otherwise cause severe moire and brightnessattenuation. Only two polarizing films 211, 213 are used, one on the topand bottom of the multi-layer stack. This creates a polarization fielddisplay, wherein each spatial light modulator comprises a liquid crystallayer functioning as a spatially-addressable voltage-controlledpolarization rotator.

In exemplary implementations, such displays are controlled so thepolarization field incident on the front polarizer (closest to theobserver) accurately reproduces the target light field.

FIG. 3 is a diagram of a polarization field display, in an exemplaryimplementation of this invention. A K-layer display is constructed byseparating multiple liquid crystal panels 301, 303, 305. The stack ofpanels is surrounded by two crossed polarizers: a front polarizer 311and a rear polarizer 313. The light field l₀(u,a) emitted by thebacklight 321 is linearly polarized by the rear polarizer 313. Thepolarization state of ray (u,a) is rotated by φ_(k)(ξ) after passagethrough layer k, where ξ=u+(d_(k)|d_(r))a. The emitted light field{tilde over (l)}(u,a) is given by applying Equation 2 to the emittedpolarization field {tilde over (θ)}(u,a) upon passage through the frontpolarizer 311.

The following is an analysis of light passing through a K layer stack ofpolarization rotators. For ease of discussion, this analysis is inflatland, considering 1D layers and 2D light fields, with a directextension to 2D layers and 4D light fields. As shown in FIG. 3, considera display of width w and height h, with K layers distributed along they-axis such that d_(k)Σ[−h/2,h/2]. Use a two-plane light fieldparameterization l(u,a). The u-axis is coincident with the x-axis andthe slope of ray (u,a) is defined as a=s−u=d_(r)tan(α), where the s-axisis a distance d_(r) from the u-axis.

The emitted light field l(u,a) is given by applying Equation 1 to thepolarization field θ(u,a) incident on the front polarizer (closest tothe observer):

l(u,a)=l ₀(u,a)sin²(θ(u,a))   (Eq. 2)

where l₀(u,a) is the light field produced by the backlight afterattenuation by the rear polarizer. The backlight is assumed to beuniform such that l₀(u,a)=l_(max) and the light field is normalized suchthat l(u,a)Σ[0,l_(max)]. This expression is used to solve for the targetpolarization field θ(u,a), as follows.

$\begin{matrix}{{\theta \left( {u,a} \right)} = {{\pm {\sin^{- 1}\left( \sqrt{\frac{l\left( {u,a} \right)}{l_{0}\left( {u,a} \right)}} \right)}}{mod}\mspace{14mu} \pi}} & \left( {{Eq}.\mspace{14mu} 3} \right)\end{matrix}$

Under these assumptions, the principal value of the arcsine ranges over[0,π/2]. Note, with full generality, the target polarization field ismulti-valued and periodic, since a rotation of ±θmodπ radians willproduce an identical intensity by application of Malus' law.

Each layer controls the spatially-varying polarization state rotationφ_(k)(ξ), as induced at point ξ along layer k. Ray (u,a) intersects theK layers, accumulating incremental rotations at each intersection, suchthat the emitted polarization field {tilde over (θ)}(u, a) is given by

$\begin{matrix}{{\overset{\sim}{\theta}\left( {u,a} \right)} = {\sum\limits_{k = 1}^{K}\; {\varphi_{k}\left( {u + {\left( {d_{k}/d_{r}} \right)a}} \right)}}} & \left( {{Eq}.\mspace{14mu} 4} \right)\end{matrix}$

Combining Equations 2 and 4 yields the following model for the lightfield {tilde over (l)}(u,a) emitted by a K-layer polarization fielddisplay:

$\begin{matrix}{{\overset{\sim}{l}\left( {u,a} \right)} = {{l_{0}\left( {u,a} \right)}{\sin^{2}\left( {\sum\limits_{k = 1}^{K}\; {\varphi_{k}\left( {u + {\left( {d_{k}/d_{r}} \right)a}} \right)}} \right)}}} & \left( {{Eq}.\mspace{14mu} 5} \right)\end{matrix}$

In exemplary implementations, the voltage applied to the respectivepixels in a multi-layer LCDs is optimized for polarization fielddisplay. In particular, when rendering a 3D scene, the voltage appliedto the respective pixels is optimized so that the light fieldtransmitted from the front polarizer in the stack (closest to theobserver) approximates the light field that would be transmitted(through the area of the screen if the screen were absent) by the actual3D scene.

Consider a discrete parameterization for which the emitted polarizationfield is represented as a column vector {tilde over (θ)} with Melements, each of which corresponds to the angle of polarization for aspecific light field ray. Similarly, the polarization state rotationsare represented as a column vector φ with N elements, each of whichcorresponds to a specific display pixel in a given layer. Under thisparameterization, Equation 4 yields a linear model such that

$\begin{matrix}{{\overset{\sim}{\theta}}_{m} = {\sum\limits_{n = 1}^{N}\; {P_{mn}\varphi_{n}}}} & \left( {{Eq}.\mspace{14mu} 6} \right)\end{matrix}$

where {tilde over (θ)}_(m) and φ_(n) denote ray m and pixel n of {tildeover (θ)} and φ, respectively. An element P_(mn) of the projectionmatrix P is given by the normalized area of overlap between pixel n andray m, occupying a finite region determined by the sample spacing.

In exemplary implementations of this invention, an optimal set ofpolarization state rotations φ is found by solving the followingconstrained linear least-squares problem:

$\begin{matrix}{{\underset{\varphi}{\arg \mspace{11mu} \min}{{\theta - {P\; \varphi}}}^{2}},{{{for}\mspace{14mu} \varphi_{\min}} \leq \varphi \leq \varphi_{\max}}} & \left( {{Eq}.\mspace{14mu} 7} \right)\end{matrix}$

where each layer can apply a rotation ranging over [φ_(min),φ_(max)].Equation 7 can be solved by a variety of techniques, including (1) asparse, constrained, large-scale trust region method or (2) asimultaneous algebraic reconstruction technique (SART). However, theproblem can be solved more efficiently by SART than by the trust regionapproach.

In exemplary implementations of this invention, SART provides aniterative solution wherein the estimate θ^((q)) at iteration q is givenby

φ^((q))=φ^((q−1)) +v°(P ^(T)(w°(θ−Pφ ^((q−1))))),   (Eq. 8)

where ° denotes the Hadamard product for element-wise multiplication andelements of the w and v vectors are given by

$\begin{matrix}{{w_{m} = \frac{1}{\sum\limits_{n = 1}^{N}\; P_{mn}}}{and}{v_{n} = \frac{1}{\sum\limits_{m = 1}^{M}\; P_{mn}}}} & \left( {{Eq}.\mspace{14mu} 9} \right)\end{matrix}$

After each iteration, additional constraints on φ^((q)) are enforced byclamping the result to the feasible rotation range. SART rapidlyconverges to a solution approaching the fidelity of that produced byalternative iterative methods, including trust region and conjugategradient descent techniques. SART allows for real-time optimization forinteractive polarization field displays.

According to principles of this invention, polarization fields presentboth an optically and computationally efficient architecture for dynamiclight field display using multi-layer LCDs.

In alternate implementations of this invention, SART is be used tocontrol a multi-layered, attenuation-based display. Theattenuation-based display is fabricated by placing a polarizer on thebacklight and additional polarizers after each liquid crystal layer,effectively creating a set of dynamically-programmable transparencies.SART is applied to attenuation layers by substituting the logarithm ofthe emitted light field intensity {tilde over (l)}_(m) the logarithm ofthe transmittance t_(n) for {tilde over (θ)}_(m) and φ_(n) in Equation6, respectively.

In a prototype of this invention, monochromatic layers and fieldsequential color are employed. Each layer of this prototype comprises amodified Barco® E-2320 PA LCD, supporting a 1600×1200 8-bit grayscaledisplay at 60 Hz, and an active area of 40.8×30.6 cm. In this prototype,the liquid crystal layer is separated from the case, backlight, andpower supply. To modify the Barco® LCD, polarizing films were removedand the adhesive was dissolved with acetone. The Barco® LCD was alsomodified by constructing an extended ribbon cable that allows thedriverboard to be folded above the display using a pair of 20-pinconnectors and a flat flexible cable. (This modification to the Barco®LCD was desirable because, in the unmodified Barco® LCD, the driverboard is folded behind the panel, and would have blocked a portion ofthe display when used in the stacked configuration of this prototype)

In this prototype, the exposed panel, driver boards, and power supplyare mounted to a waterjet-cut aluminum frame. Four such panels wereconstructed and stacked on a wooden stand. Arbitrary layer spacings aresupported by translating the frames along rails. Acrylic spacers holdthe layers at a fixed spacing of 1.7 cm for all experiments described inthis paper, yielding a total display thickness of 5.1 cm. The prototypeis illuminated using an interleaved pair of backlights and controlled bya 3.4 GHz Intel Core® i7 workstation with 4 GB of RAM. A four-headNVIDIA® Quadro® NVS 450 graphics card synchronizes the displays.

This prototype operates in either attenuation-based orpolarization-based modes. The original polarizers were discarded andreplaced with AP38-006T linear polarizers (obtained from AmericanPolarizers, Inc., Reading, Pa.). By specification, a single polarizerhas a transmission efficiency of 38% for unpolarized illumination.Transmission is reduced to 30% through a pair of aligned polarizers,yielding an efficiency of 80% for polarized light passing through asingle, aligned polarizer. Five polarizers are required forattenuation-based display, with a pair of crossed polarizers on the rearlayer followed by successively-crossed polarizers on each remaininglayer. A polarization field display is implemented by enclosing thestack by a single pair of crossed polarizers.

In this prototype, field sequential color is simulated, for stillimagery, by combining three photographs taken while alternating thecolor channel displayed on each layer.

In this prototype, each panel is radiometrically calibrated to allow anaccurate mapping from optimized rotation angles to displayed imagevalues. The Barco® E-2320 PA is intended for medical diagnostic imagingand replicates the DICOM (Digital Imaging and Communications inMedicine) Grayscale Standard Display Function. The normalized displayedintensity IΣ[0,1] was measured as a function of the 8-bit image valuevΣ[0,255] using a photometer held against an unmodified panel. Theresulting radiometric response curve is approximated by a gamma value ofγ=3.5 such that I=(v/255)^(γ). Thus, gamma compression maps optimizedpixel transmittances to image values when operating in theattenuation-based mode. When operated as a polarization field display,optimization yields the polarization state rotation φ for each pixel.For an LCD panel enclosed by two polarizers (e.g., the originalunmodified BARCO panel), this mapping is modeled by Equation 1 such thatI=sin²(φ). Equating this with the gamma curve yields the followingmapping between rotations and image values.

v(φ)=└255 sin^(2/γ)(φ)+0.5┘  (Eq. 10)

In this prototype, the light fields may be rendered with a spatialresolution of 512×384 pixels and may depict 3D scenes with bothhorizontal and vertical parallax from 7×7 viewpoints within a field ofview of 10 degrees. POV-Ray (Persistence of Vision Raytracer) is used torender the scenes. During the process of computing the rendering, a 4Dantialiasing filter is applied to the light fields by rendering eachview with a limited depth of field. This antialiasing filtersimultaneously approximates the limited depth of field established formulti-layer light field displays.

In this prototype, Equation 7 is solved by using either a large-scaletrust region or SART.

For example, this prototype can employ a Matlab® LSQLIN solver to solveEquation 7, implementing a sparse, constrained, large-scale trust regionmethod. This solver converges in about 8 to 14 iterations for three tofive attenuating or polarization-rotating layers. Solutions are foundwithin approximately 10 minutes on the previously-described Intel Core®i7 workstation.

Or, for example, this prototype can implement the SART algorithm givenby Equation 8, using Matlab® and on the GPU (graphical processing unit).Advantageously, SART is well-suited for parallel processing onprogrammable GPUs. In the prototype, the software code is programmed inC++, OpenGL, and Cg (C for Graphics). Light fields are rendered andantialiased in real-time using OpenGL, followed by several iterations ofthe GPU-based SART implementation. Using SART, this prototype achievesrefresh rates of up to 24 frames per second using one iteration for fourlayers running on the NVIDIA® Quadro® NVS 450. In a trial of theprototype, the SART converged rapidly, with 2 to 5 iterations minimizingreconstruction artifacts. Estimates for the previous frame may seed theoptimization for the current frame. For static scenes, this effectivelyimplements an increasing number of SART iterations over time, whileproviding a suitable initialization for successive frames in a dynamicenvironment.

This prototype achieves automultiscopic display. However, this prototypeexhibits artifacts not predicted by simulations. Moire is present,although it could be mitigated. Without being limited by theory, itappears that: (a) intensity artifacts arise from discrepancies betweenthe prototype and ideal construction using polarization-rotating layers(including, apparently, the presence of multiple liquid crystal domainsin the panels); and (b) the commercial panels used in the prototype donot operate precisely as two-dimensional polarization rotators,particularly at oblique angles.

In a trial of this prototype, a photometer measured the normalizedintensity l as differing image values v₁ and v₂ were displayed on therear and front layer, respectively. Substituting Equation 10 intoEquation 5 yields the following prediction.

$\begin{matrix}{{I\left( {v_{1},v_{2}} \right)} = {\sin^{2}\left\{ {{\sin^{- 1}\left\lbrack \left( \frac{v_{1}}{255} \right)^{\frac{\gamma}{2}} \right\rbrack} + {\sin^{- 1}\left\lbrack \left( \frac{v_{2}}{255} \right)^{\frac{\gamma}{2}} \right\rbrack}} \right\}}} & \left( {{Eq}.\mspace{14mu} 11} \right)\end{matrix}$

In this trial of this prototype, measured intensities were nearlyidentical to this prediction (upon interchanging v₁ and v₂), validatingthe additive model in Equation 4. Measured contrast is limited when v₁and v₂ are large. Visual artifacts persist in this prototype. Withoutbeing limited by theory, these artifacts appear to be due to differencesbetween off-the-shelf panels and ideal polarization rotators.

Polarization fields can accurately present objects beyond the display,but can also be operated in a volumetric mode enclosing the scene forreduced errors.

In trials, polarization fields (in a prototype of this invention)perform comparably to attenuation layers in terms of reconstructionfidelity. Yet, halo artifacts are noticeably reduced. Without beinglimited by theory, this may be attributable primarily to differentbiases introduced by least-squares optimization of transformed objectivefunctions. Attenuation-based displays optimize an objective, reminiscentof Equation 7, defined for the logarithm of the target intensities. Thispenalizes artifacts in dark regions, leading to the observed halos. Bycomparison, polarization fields (in a prototype of this invention)optimize an objective defined for target intensities transformed byEquation 3; this transformation is more linear than for attenuation,thereby mitigating halos.

In exemplary implementations of this invention, polarization fields haveseveral notable benefits over existing automultiscopic displays,particularly those supporting relatively thin form factors. Polarizationfields compare favorably to attenuation layers: among other things,polarization fields may have increased optical efficiency and reducedreconstruction artifacts, as compared to attenuation layers. Bothmethods significantly improve upon parallax barriers and integralimaging, presenting imagery with greater spatial resolution, increasedbrightness, and extended depth of field.

Artifacts observed in a prototype of this invention are not predicted bythe polarization rotator model. Without being limited by theory, itappears that the artifacts can be primarily attributed to the presenceof multiple liquid crystal domains in the in-plane switching (IPS)panels used in the prototype. According to principles of this invention,a multi-domain LCD model that employs Jones calculus may account for theartifacts.

The Jones matrix modeling an LCD depends on its architecture. Yet, allLCDs are fundamentally retardation-based and can be approximated asrotated half-wave plates, with Jones matrix:

$\begin{matrix}{{J_{HWP}(\alpha)} = \begin{pmatrix}{\cos \left( {2\alpha} \right)} & {\sin \left( {2\alpha} \right)} \\{\sin \left( {2\alpha} \right)} & {- {\cos \left( {2\alpha} \right)}}\end{pmatrix}} & \left( {{Eq}.\mspace{14mu} 12} \right)\end{matrix}$

where α is the liquid crystal director angle.

Compared to a true polarization rotator, each LCD acts as apseudo-rotator: reversing the polarization state and doubling therotation angle. The following expression models the normalized intensityfor K-layer compositions of single-domain LCDs enclosed by crossedlinear polarizers:

$\begin{matrix}\begin{matrix}{{I_{{HWP} - K - 1}(\alpha)} = {I_{0}{{\left( {0\mspace{14mu} 1} \right)\left( {\prod\limits_{k = 1}^{K}\; {J_{HWP}\left( \alpha_{K - k + 1} \right)}} \right)\begin{pmatrix}1 \\0\end{pmatrix}}}^{2}}} \\{= {I_{0}{\sin^{2}\left( {\sum\limits_{k = 1}^{K}\; {\left( {- 1} \right)^{k - 1}2\alpha_{k}}} \right)}}}\end{matrix} & \left( {{Eq}.\mspace{14mu} 13} \right)\end{matrix}$

For the choice α_(k)=(−1)^(k−1)φ_(k)/2, this expression is identical toEquation 5. Thus, under this model, multi-layer, single-domain LCDs canapproximate layered polarization rotators.

Assume every IPS pixel is divided into two domains. Each domain i inlayer k is approximated as a rotated half-wave plate J_(HWP)(α_(k)^((i))) with symmetric directors such that α_(k) ⁽¹⁾=−α_(k) ⁽²⁾=α. Whenthe angle between the linear polarizers is not a multiple of 90 degrees,the normalized intensity for a single multi-domain panel differs fromEquation 13.

For a multi-layer, multi-domain LCD, rays emitted by the backlight willpass through a single domain in each layer. Considering a bundle of rayspassing through a local region, the intensity will depend on theweighted average due to passing through all domain combinations. Summingover combinations yields the following expression for normalizedintensity for two-layer, two-domain LCDs.

$\begin{matrix}\begin{matrix}{{{I_{{HWP} - 2 - 2}\left( {\alpha_{1},\alpha_{2}} \right)} = {\frac{I_{0}}{4}{\sum\limits_{i = 1}^{2}\; \sum\limits_{j = 1}^{2}}}}\;} \\{{{\left( {0\mspace{20mu} 1} \right){J_{HWP}\left( \alpha_{2}^{(j)} \right)}{J_{HWP}\left( \alpha_{1}^{(i)} \right)}\begin{pmatrix}1 \\0\end{pmatrix}}}^{2}} \\{= {I_{0}\left( \frac{{\sin^{2}\left( {2\left( {\alpha_{1} + \alpha_{2}} \right)} \right)} + {\sin^{2}\left( {2\left( {\alpha_{1} - \alpha_{2}} \right)} \right)}}{2} \right)}}\end{matrix} & \left( {{Eq}.\mspace{14mu} 14} \right)\end{matrix}$

Without being limited by theory: (a) Equation 14 provides intuition intohow multi-layer, multi-domain LCDs deviate from polarization rotators;(b) the first term is proportional to Equation 5, whereas the secondterm constitutes the error under a polarization rotator approximation.

Extending this analysis to four layers yields the following expression.

$\begin{matrix}{{I_{{HWP} - 4 - 2}(\alpha)} = {I_{0}\left( \frac{1 - {\prod\limits_{k = 1}^{4}\; {\cos \left( {4\alpha_{k}} \right)}}}{2} \right)}} & \left( {{Eq}.\mspace{14mu} 15} \right)\end{matrix}$

Also, extending the analysis to four-layer LCDs yields the followingexpression for the normalized intensity I_(HWP-4-2)(α₁,α₂,α₃,α₄) whenenclosing such displays with two crossed linear polarizers, one at eachend of the stack:

$\begin{matrix}{{I_{{HWP} - 4 - 2}\left( {\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}} \right)} = \mspace{169mu} {{\frac{I_{0}}{16}{\sum\limits_{i = 1}^{2}\; {\sum\limits_{j = 1}^{2}\; {\sum\limits_{k = 1}^{2}\; {\sum\limits_{l = 1}^{2}\; \mspace{31mu} {{\left( {0\mspace{20mu} 1} \right){J_{HWP}\left( \alpha_{4}^{(l)} \right)}{J_{HWP}\left( \alpha_{3}^{(k)} \right)}{J_{HWP}\left( \alpha_{2}^{(j)} \right)}{J_{HWP}\left( \alpha_{1}^{(i)} \right)}\begin{pmatrix}1 \\0\end{pmatrix}}}^{2}}}}}} = {\frac{I_{0}}{8}\left\lbrack {{\sin^{2}\left( {2\left( {\alpha_{1} + \alpha_{2} + \alpha_{3} + \alpha_{4}} \right)} \right)} + {\sin^{2}\left( {2\left( {{- \alpha_{1}} + \alpha_{2} + \alpha_{3} + \alpha_{4}} \right)} \right)} + \mspace{20mu} {{\sin^{2}\left( {2\left( {\alpha_{1} - \alpha_{2} + \alpha_{3} + \alpha_{4}} \right)} \right)} {\quad{{+ \sin^{2}}{\quad {\left( {2\left( {\alpha_{1} + \alpha_{2} - \alpha_{3} + \alpha_{4}} \right)} \right) + \left. \quad \mspace{14mu} {{\sin^{2}\left( {2\left( {\alpha_{1} + \alpha_{2} + \alpha_{3} - \alpha_{4}} \right)} \right)} + {\sin^{2}\left( {2\left( {\alpha_{1} - \alpha_{2} - \alpha_{3} + \alpha_{4}} \right)} \right)} +  {\sin^{2}\left( {2\left( {\alpha_{1} - \alpha_{2} + \alpha_{3} - \alpha_{4}} \right)} \right)} + {\sin^{2}\left( {2\left( {\alpha_{1} + \alpha_{2} - \alpha_{3} - \alpha_{4}} \right)} \right)}} \right\rbrack}}}}}} \right.}}} & \left( {{Eq}.\mspace{14mu} 16} \right)\end{matrix}$

As before, it is assumed the domains are symmetrically oriented on eachlayer (i.e., α_(k) ⁽¹⁾=−α_(k) ⁽²⁾=α_(k)). In practice, it is desirablefor each term to be weighted by the likelihood of traversing thecorresponding combination of domains (depending on the geometricarrangement and scattering properties of the panels). Equation 16assumes equal weighting. Without being limited by theory: (a) the firstterm in Equation 16 corresponds to a desirable multi-layer polarizationrotator model, and (b) the remaining terms in Equation 16 constituteartifacts introduced by multi-domain LCD panels, compared to the desiredimplementation. Equation 16 accurately predicts the artifacts exhibitedby the prototype.

The polarization rotator approximation deviates from both trials of aprototype of this invention and the multi-domain model (particularly forlarge image values). For small image values or cases for which valuesare large for a single layer, measurements and the multi-domain modelare well approximated.

Without being limited by theory, it appears that the presence ofmultiple domains is the primary source of artifacts in a prototype ofthis invention. This insight reveals solutions. Since the multi-domainmodel accurately predicts experimental artifacts, it may be used for anenhanced optimization procedure. However, Equation 15 is non-linear andnot directly amenable to real-time optimization via the SART algorithm.Alternatively, replacing panels with single-domain alternatives betterapproximates polarization rotators (as predicted by Equation 13). Inpractice, both strategies may be pursued, together with laboratorycharacterizations, to obtain better performance of polarization fielddisplays.

In a prototype of this invention, software that employs SART isprogrammed in C++ and uses OpenGL and Cg shaders. The algorithm assumesa light field with N distinct views and K layers positioned atuser-defined depths along the optical axis. Intermediate quantities,including the target light field views and temporary variables storingweights and layer patterns, are internally rendered into 16-bit off-lineframebuffers (e.g., framebuffer objects (FBOs)) before the optimizedpatterns are displayed on the individual polarization-rotating layers.Only three separate Cg fragment programs are used, each performing theaction implied by their names.

FIG. 4 is a flow chart for controlling a stack of polarization rotatorswith a linear optimization algorithm. As shown in FIG. 4, a processormay perform a linear optimization algorithm 401 to optimize rotationpolarization states at respective pixels in the stack, in order torender a light field that best approximates a target light field. Thelinear approximation algorithm 401 may process either (a) capturedmulti-view or light field data 403, or (b) synthetic multi-view or lightfield data 405. Optionally, the processor may use a 3D model 407 inorder to create the synthetic data 405. The design of the linearoptimization algorithm takes into account hardware specifications of themulti-layer LCD stack, with simplifying assumptions 409. Using thelinear optimization algorithm 401, the processor outputs polarizationrotation mask patterns 411 or control signals to achieve these patterns.The state of the polarization rotator stack is accordingly changed 413.A human viewer looking at the display perceives a 3D display 415.

FIG. 5 is a flow chart for controlling a stack of polarization rotatorswith a non-linear optimization algorithm (rather than a linearoptimization program). As shown in FIG. 5, a processor may perform anon-linear optimization algorithm 501 to optimize rotation polarizationstates at respective pixels in the stack, in order to render a lightfield that best approximates a target light field. The non-linearapproximation algorithm 501 may process either (a) captured multi-viewor light field data 503, or (b) synthetic multi-view or light field data505. Optionally, the processor may use a 3D model 507 in order to createthe synthetic data 505. The design of the non-linear optimizationalgorithm takes into account hardware specifications of the multi-layerLCD stack 509. Using the non-linear optimization algorithm 501, theprocessor outputs polarization rotation mask patterns 511 or controlsignals to achieve these patterns. The state of the polarization rotatorstack is accordingly changed 513. A human viewer looking at the displayperceives a 3D display 515.

In the examples shown in FIGS. 4 and 5, the optimization calculation401, 501 may compute an optimal set of polarization state rotationsinduced in light at respective pixels in the polarization rotators. Foreach respective light ray in a set of light rays, the optimizationcalculation 401, 501 may include computing a summation of changes to thepolarization state rotation of the respective light ray that occur asthe respective light ray travels through the stack of polarizationrotators. Also, for example, an optimization calculation 401 may includecomputing, according to Malus' law, an intensity of the respective lightray, the intensity being as of when the respective light ray emergesfrom the polarizer. Malus' law includes a term that is a square of asinusoidal function. An optimization calculation 401 may include solvingfor the argument of the sinusoidal function, using an approximationbased on only a single period of the sinusoidal function.

In exemplary implementations of this invention, light emerging from thefront polarizer produces an automultiscopic display.

Definitions and Clarifications:

Here are a few definitions and clarifications. As used herein:

The terms “a” and “an”, when modifying a noun, do not imply that onlyone of the noun exists.

An “automultiscopic” display is a display by a flat screen device of a3D image, which display, when viewed by a human not wearing glasses orother optical apparatus: (a) exhibits motion parallax and binocularparallax, and (b) includes multiple views, the view seen depending onthe angle at which the image is viewed.

The term “comprise” (and grammatical variations thereof) shall beconstrued broadly, as if followed by “without limitation”. If Acomprises B, then A includes B and may include other things.

The term “e.g.” means including without limitation.

To minimize the “error” between two things is to minimize a measure ofor based on the difference between the two things. For example, asolution to a linear least squares problem may minimize the errorbetween an actual and a target light field.

The fact that an “example” or multiple examples of something are givendoes not imply that they are the only instances of that thing. Anexample (or a group of examples) is merely a non-exhaustive andnon-limiting illustration.

In the context of a display device (and components of the device),“front” is optically closer to a viewer, and “rear” is optically furtherfrom the viewer, when the viewer is viewing a display produced by thedevice during normal operation of the device. The “front” and “rear” ofa display device continue to be the front and rear, even when no vieweris present. A stack of polarization rotators is a display device or acomponent of a display device; thus, a stack of polarization rotatorshas a “front” and a “rear”.

The term “include” (and grammatical variations thereof) shall beconstrued broadly, as if followed by “without limitation”.

Intensity” shall be construed broadly to include any measure of orrelated to intensity, energy or power. For example, the “intensity” oflight includes any of the following measures: irradiance, spectralirradiance, radiant energy, radiant flux, spectral power, radiantintensity, spectral intensity, radiance, spectral radiance, radiantexitance, radiant emittance, spectral radiant exitance, spectral radiantemittance, radiosity, radiant exposure and radiant energy density.

The term “Malus” law” shall be construed broadly to include anyformulation of that law and any computation equivalent to that law. Forexample, the term “Malus' law” includes a sine squared version and acosine squared version of that law. Equation 1 above (I=I₀sin²(θ)) is aversion of Malus' law.

The term “or” is an inclusive disjunctive. For example “A or B” is trueif A is true, or B is true, or both A or B are true.

“Parallax” includes binocular parallax and motion parallax. A displayexhibits binocular parallax, if the apparent position of an objectviewed by the left eye and the right eye of a human viewer differsbecause of the different positions of the two eyes. A display exhibitsmotion parallax, if the apparent position of an object appears to changeas the viewpoint of the human viewer moves (e.g., by moving the viewer'shead).

A parenthesis is simply to make text easier to read, by indicating agrouping of words. A parenthesis does not mean that the parentheticalmaterial is optional or can be ignored.

To vary something “per pixel” means to vary it at respective pixels.

A “pixel” includes the smallest addressable element in a display device.For example, a light-transmitting or light-emitting display device mayhave pixels.

A “polarization rotator” is a device configured to change thepolarization state of light that travels through the device. Forexample, a polarization rotator may comprise a layer of liquid crystalbetween a pair of transparent electrodes. Or, for example, any devicethat alters the polarization state rotation of light passing through thedevice is a polarization rotator.

The term “polarization state rotation” shall be construed broadly. Forexample, the term includes a rotation of the angle of polarization oflinearly polarized light.

The term “polarizer” means a device that filters light according to thelight's polarization state. For example, a polarizing diffuser is a“polarizer”.

A “sinusoidal” function includes a sine function and a cosine function.

“SART” means simultaneous algebraic reconstruction technique.

Variations:

This invention may be implemented in many different ways. Here are somenon-limiting examples.

In exemplary implementations of this invention, additional elements maybe included to mitigate moire, scattering, and reflections.

Laboratory measurement of the Jones matrices characterizing the displaypanels, together with a modified image formation model, may be used tominimize artifacts.

Holographic diffusers may be included to mitigate moire. Fresnel lensesmay be added to extend the depth of field.

Conventional LCD panels can be replaced with displays that behave aspolarization rotators. Alternatively, additional optical elements (e.g.,compensation films) may be added to produce a similar result. Forexample, panels 201, 203, 205, 207 in FIG. 2 may comprise displays thatbehave as polarization rotators, or may include additional opticalelements, such as compensation films, to produce a similar result.

A frequency-domain analysis of polarization fields may yield an analyticdepth of field expression.

Alternatively, to expand the degrees of freedom, time-multiplexed,multi-layer decompositions may be used.

Consider the multi-valued, periodic target polarization field given byEquation 3. In a prototype implementation, only the principal value ofthis expression is considered, limiting the target polarization field toθ(u,a)Σ[0,π/2]. If this restriction is lifted, additional degrees offreedom are accessible; for example, larger rotations can decrease theintensity of an emitted ray via application of Malus' law.

Panels can be used that apply positive and negative rotations over thefull range such that φ_(k)(ξ)Σ[−π,π], where ξ=u+(d_(k)/d_(r))a (see FIG.3). This may increase reconstruction fidelity and may enable efficient,unconstrained optimization methods.

This invention may be implemented as a method comprising, incombination: (a) using an illumination source to provide light that istransmitted through at least (i) a stack of polarization rotators and(ii) a polarizer; and (b) using one or more processors (i) to perform anoptimization calculation to compute a set of polarization staterotations induced in the light at respective pixels of the polarizationrotators; and (ii) to output control signals to control the polarizationstate rotations induced in the light at the respective pixels; wherein(I) each of the polarization rotators is a layer in the stack, (II) thepolarizer is optically in front of the stack, (III) each of thepolarization rotators comprises a spatially addressable device, thedevice being configured to dynamically vary per pixel polarization staterotations induced in the light, and (IV) for each respective light rayin a set of light rays, the optimization calculation includes computinga summation of changes to polarization state rotation of the respectivelight ray that occur as the respective light ray travels through thestack of polarization rotators. Furthermore: (1) the optimizationcalculation may include computing, according to Malus' law, an intensityof the respective light ray, the intensity being as of when therespective light ray emerges from the polarizer; (2) Malus' law includesa term that is a square of a sinusoidal function, the sinusoidalfunction has an argument, and the optimization calculation may includesolving for the argument, using an approximation based on only a singleperiod of the sinusoidal function; (3) each of the polarization rotatorsmay comprise a layer of liquid crystal; (4) each of the polarizationrotators may be monochromatic and the illumination source may be astrobe backlight configured to sequentially illuminate the polarizationrotators with varying colors of light; (5) each of the polarizationrotators, respectively, may be configured to dynamically vary voltageapplied at the respective pixels in order to control polarization staterotation induced in the light at the respective pixels; (6) theoptimization calculation may be a constrained linear least squaresoptimization calculation; (7) the one or more processors may beconfigured to employ a SART technique when performing the optimizationcalculation; (8) the set of polarization state rotations, which may becomputed by the optimization calculation, may minimize error between alight field transmitted from the polarizer and a light field that wouldbe created by a target 3D scene; and (9) light emerging from thepolarizer may produce an automultiscopic display.

In some implementations of this invention: (a) for a least one pair ofadjacent polarization rotators in the stack, no polarizer is positionedbetween the pair; or (b) the optimization calculation performed by theone or more processors does not perform operations on values that areindicative of per pixel attenuation of the light and that are for pixelsin a polarization rotator other than the front polarization rotator inthe stack.

This invention may be implemented as apparatus comprising, incombination: (1) a stack of polarization rotators, each of thepolarization rotators being a layer in the stack; (2) a polarizer, thepolarizer being optically in front of the stack; (3) an illuminationsource, the illumination source being configured to provide light thatis transmitted through at least the stack and the polarizer; and (4) oneor more processors; wherein (a) the one or more processors areconfigured (i) to perform an optimization calculation to compute a setof polarization state rotations induced in the light at respectivepixels of the polarization rotators, and (ii) to output control signalsto control the polarization state rotations induced in the light at therespective pixels, (b) each of the polarization rotators comprises aspatially addressable device, the device being configured to dynamicallyvary per pixel polarization state rotations induced in the light, and(c) for each respective light ray in a set of light rays, theoptimization calculation includes computing a summation of changes topolarization state rotation of the respective light ray that occur asthe respective light ray travels through the stack of polarizationrotators. Furthermore: (1) the optimization calculation may includecomputing, according to Malus' law, an intensity of the respective lightray, the intensity being as of when the respective light ray emergesfrom the polarizer; (2) Malus' law includes a term that is a square of asinusoidal function, the sinusoidal function has an argument, and theoptimization calculation may include solving for the argument, using anapproximation based on only a single period of the sinusoidal function;(2) the set of polarization state rotations, which may be computed bythe optimization calculation, may minimize error between a light fieldtransmitted from the polarizer and a light field that would be createdby a target 3D scene; (3) each of the polarization rotators may bemonochromatic and the illumination source may be a strobe backlightconfigured to sequentially illuminate the polarization rotators withvarying colors of light; and (4) the optimization calculation may be alinear optimization calculation and the one or more processors may beconfigured to employ a SART technique when performing the linearoptimization calculation.

It is to be understood that the methods and apparatus that are describedabove and below are merely illustrative applications of the principlesof the invention. Numerous modifications may be made by those skilled inthe art without departing from the scope of the invention.

The following is a computer programming listing (in pseudocode) for analgorithm that employs SART. The algorithm is used in an illustrativeimplementation of this invention:

Computer Programming Listing:

variables FBO_LF[N], FBO_LF_TMP[N], FBO_LAYERS[K], FBO_W[N], FBO_V[K],DEPTH[K] function displayLoop( )    if not initialized       initializeall FBO_W, FBO_V    end    drawLightField( );    runSART( );   drawReconstructedLayers( ); end function drawLightField( )    for alllight field views i       activate FBO_LF[i]       set perspective i      drawScene( ); // render desired 3D scene (e.g., a teapot)    endend function runSART( )    for all iterations k    // 1. computeAx^((k))       for all light field views i          activateFBO_LF_TMP[i]          enable BLEND_MODE          set perspective i         for all layers l             draw 2D plane at DEPTH[l] texturedwith             FBO_LAYERS[l]          end       end       // 2. givenAx^((k)), compute W (b − Ax^((k)))       for all light field views i         activate FBO LF TMP[i]          activateCG_SHADER_MULTIPLY_SUBTRACT(FBO_W[i],          FBO_LF[i],FBO_LF_TMP[i])         draw orthographic 2D plane with normalized texcoords       end      // 3. given W (b − Ax^((k))), compute VA^(T) (W (b − Ax^((k))))      for all layers l          activate FBO_LAYER[l]          activateBLEND_MODE          for all light field views i             setprojective texture to perspective i             activateCG_SHADER_MULTIPLY(FBO_V[l],FBO             LAYER[l])           draworthographic 2D plane with automatic texcoord generation          end      end       // 4. enforce constraints by clamping values outsidefeasible range       for all layers l          activate FBO_LAYER[l]         activate CG_SHADER_CLAMP(FBO_LAYER[l])          draworthographic 2D plane textured with FBO_LAYER[l]       end    end endfunction draw ReconstructedLayers( )    for all layers l       setviewport for display 1       draw orthographic 2D plane textured withFBO_LAYERS[l]    end end

What is claimed is:
 1. A method comprising, in combination: (a) using anillumination source to provide light that is transmitted through atleast (i) a stack of polarization rotators and (ii) a polarizer; and (b)using one or more processors (i) to perform an optimization calculationto compute a set of polarization state rotations induced in the light atrespective pixels of the polarization rotators; and (ii) to outputcontrol signals to control the polarization state rotations induced inthe light at the respective pixels; wherein (I) each of the polarizationrotators is a layer in the stack, (II) the polarizer is optically infront of the stack, (III) each of the polarization rotators comprises aspatially addressable device, the device being configured to dynamicallyvary per pixel polarization state rotations induced in the light, and(IV) for each respective light ray in a set of light rays, theoptimization calculation includes computing a summation of changes topolarization state rotation of the respective light ray that occur asthe respective light ray travels through the stack of polarizationrotators.
 2. The method of claim 1, wherein the optimization calculationincludes computing, according to Malus' law, an intensity of therespective light ray, the intensity being as of when the respectivelight ray emerges from the polarizer.
 3. The method of claim 2, wherein(a) Malus' law includes a term that is a square of a sinusoidalfunction; (c) the sinusoidal function has an argument; (c) theoptimization calculation includes solving for the argument, using anapproximation based on only a single period of the sinusoidal function.4. The method of claim 1, wherein for a least one pair of adjacentpolarization rotators in the stack, no polarizer is positioned betweenthe pair.
 5. The method of claim 1, wherein the optimization calculationdoes not perform operations on values that are indicative of per pixelattenuation of the light and that are for pixels in a polarizationrotator other than the front polarization rotator in the stack.
 6. Themethod of claim 1, wherein each of the polarization rotators comprises alayer of liquid crystal.
 7. The method of claim 1, wherein each of thepolarization rotators is monochromatic and the illumination source is astrobe backlight configured to sequentially illuminate the polarizationrotators with varying colors of light.
 8. The method of claim 1, whereineach of the polarization rotators, respectively, is configured todynamically vary voltage applied at the respective pixels in order tocontrol polarization state rotation induced in the light at therespective pixels.
 9. The method of claim 1, wherein the optimizationcalculation is a constrained linear least squares optimizationcalculation.
 10. The method of claim 1, wherein the one or moreprocessors are configured to employ a SART technique when performing theoptimization calculation.
 11. The method of claim 1, wherein the set ofpolarization state rotations, which is computed by the optimizationcalculation, minimizes error between a light field transmitted from thepolarizer and a light field that would be created by a target 3D scene.12. The method of claim 1, wherein light emerging from the polarizerproduces an automultiscopic display.
 13. Apparatus comprising, incombination: a stack of polarization rotators, each of the polarizationrotators being a layer in the stack; a polarizer, the polarizer beingoptically in front of the stack; an illumination source, theillumination source being configured to provide light that istransmitted through at least the stack and the polarizer; and one ormore processors; wherein (a) the one or more processors are configured(i) to perform an optimization calculation to compute a set ofpolarization state rotations induced in the light at respective pixelsof the polarization rotators, and (ii) to output control signals tocontrol the polarization state rotations induced in the light at therespective pixels, (b) each of the polarization rotators comprises aspatially addressable device, the device being configured to dynamicallyvary per pixel polarization state rotations induced in the light, and(c) for each respective light ray in a set of light rays, theoptimization calculation includes computing a summation of changes topolarization state rotation of the respective light ray that occur asthe respective light ray travels through the stack of polarizationrotators.
 14. The apparatus of claim 13, wherein the optimizationcalculation includes computing, according to Malus' law, an intensity ofthe respective light ray, the intensity being as of when the respectivelight ray emerges from the polarizer.
 15. The apparatus of claim 14,wherein (a) Malus' law includes a term that is a square of a sinusoidalfunction; (c) the sinusoidal function has an argument; (c) theoptimization calculation includes solving for the argument, using anapproximation based on only a single period of the sinusoidal function.16. The apparatus of claim 13, wherein for a least one pair of adjacentpolarization rotators in the stack, no polarizer is positioned betweenthe pair.
 17. The apparatus of claim 13, wherein the optimizationcalculation does not perform operations on values that are indicative ofper pixel attenuation of the light and that are for pixels in apolarization rotator other than the front polarization rotator in thestack.
 18. The apparatus of claim 13, wherein the set of polarizationstate rotations, which is computed by the optimization calculation,minimizes error between a light field transmitted from the polarizer anda light field that would be created by a target 3D scene.
 19. Theapparatus of claim 13, wherein each of the polarization rotators ismonochromatic and the illumination source is a strobe backlightconfigured to sequentially illuminate the polarization rotators withvarying colors of light.
 20. The apparatus of claim 13, wherein theoptimization calculation is a linear optimization calculation and theone or more processors are configured to employ a SART technique whenperforming the linear optimization calculation.